Integral domains and the IDF property

TL;DR

This paper explores the properties and behaviors of IDF-domains, focusing on their ascent in polynomial rings, the special classes TIDF and PIDF, and their stability under various algebraic constructions.

Contribution

It proves that the IDF property ascends in PSP-domains, generalizing previous results, and analyzes the behavior of TIDF and PIDF domains under polynomial rings, localizations, and other constructions.

Findings

IDF property ascends in PSP-domains

TIDF and PIDF domains' behavior under polynomial rings and localizations

IDF property does not always ascend to polynomial rings in general

Abstract

An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P.~Malcolmson and F. Okoh proved that the IDF property does not ascend from integral domains to their corresponding polynomial rings, answering a question posed by D. D. Anderson, D. F. Anderson, and the second author two decades before. Here we prove that the IDF property ascends in the class of PSP-domains, generalizing the known result (also by Malcolmson and Okoh) that the IDF property ascends in the class of GCD-domains. We put special emphasis on IDF-domains where every nonunit is divisible by an irreducible, which we call TIDF-domains, and we also consider PIDF-domains, which form a special class of IDF-domains introduced by Malcolmson and Okoh in 2006. We investigate both the TIDF and the PIDF properties under taking polynomial rings and localizations. We also delve into their behavior under monoid domain and $D+M$ constructions